\subsection{分式的乘方}\label{subsec:8-5}
\begin{enhancedline}

根据乘方的意义和分式乘法法则，可得
\begin{align*}
    & \left(\dfrac{a}{b}\right)^2 = \dfrac{a}{b} \cdot \dfrac{a}{b} = \dfrac{a \cdot a}{b \cdot b} = \dfrac{a^2}{b^2} \douhao \\
    & \left(\dfrac{a}{b}\right)^3 = \dfrac{a}{b} \cdot \dfrac{a}{b} \cdot \dfrac{a}{b} = \dfrac{a \cdot a \cdot a}{b \cdot b \cdot b} = \dfrac{a^3}{b^3} \douhao \\
    & \quad \cdots\cdots\cdots \hspace*{5em} \cdots\cdots\cdots \juhao
\end{align*}

一般地，当 $n$ 为正整数时，
\begin{align*}
    \left(\dfrac{a}{b}\right)^n
        = \underbrace{\dfrac{a}{b} \cdot \dfrac{a}{b} \cdots\cdots \dfrac{a}{b}}_{n \text{个} \dfrac{a}{b}}
        = \dfrac{\overbrace{a \cdots\cdots a}^{n \text{个} a}}{\underbrace{b \cdots\cdots b}_{n \text{个} b}}
        = \dfrac{a^n}{b^n} \douhao
\end{align*}
即
\begin{center}
    \setlength{\fboxsep}{.6em}
    \framebox{\quad
        $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} \quad (n\text{为正整数})\juhao$
        \;}
\end{center}

这就是说，\zhongdian{分式乘方，把分子、分母各自乘方。}

\liti[0] 计算：
\begin{xiaoxiaotis}

    \begin{tblr}{columns={18em, colsep=0pt}}
        \xxt{$\left(\dfrac{5}{3y}\right)^2$；} & \xxt{$\left(\dfrac{3a^2b}{-2c^3}\right)^3$；} \\
        \xxt{$\left(-\dfrac{a^2}{b}\right)^2 \cdot \left(-\dfrac{b^2}{a}\right)^3 \div \left(-\dfrac{b}{a}\right)^4$。}
    \end{tblr}

\resetxxt
\jie \xxt{$\left(\dfrac{5}{3y}\right)^2 = \dfrac{5^2}{(3y)^2} = \dfrac{25}{9y^2}$；}

\xxt{$\left(\dfrac{3a^2b}{-2c^3}\right)^3 = \dfrac{(3a^2b)^3}{(-2c^3)^3} = \dfrac{27a^6b^3}{-8c^9} = -\dfrac{27a^6b^3}{8c^9}$；}

\xxt{$\left(-\dfrac{a^2}{b}\right)^2 \cdot \left(-\dfrac{b^2}{a}\right)^3 \div \left(-\dfrac{b}{a}\right)^4 = \dfrac{a^4}{b^2} \cdot \left(-\dfrac{b^6}{a^3} \cdot \dfrac{a^4}{b^4}\right) = -a^5$。}

\end{xiaoxiaotis}


\lianxi
\begin{xiaotis}

\xiaoti{（口答）计算：}
\begin{xiaoxiaotis}

    \begin{tblr}{columns={18em, colsep=0pt}}
        \xxt{$\left(\dfrac{3}{a^2}\right)^2$；} & \xxt{$\left(\dfrac{-b^3}{a}\right)^2$；} \\
        \xxt{$\left(\dfrac{2x}{-y}\right)^2$；} & \xxt{$\left(\dfrac{-x}{y}\right)^3$。}
    \end{tblr}

\end{xiaoxiaotis}

\xiaoti{计算：}
\begin{xiaoxiaotis}

    \begin{tblr}{columns={18em, colsep=0pt}}
        \xxt{$\left(\dfrac{-2x}{3y}\right)^2$；} & \xxt{$\left(\dfrac{-3x}{2y}\right)^3$；} \\
        \xxt{$\left(\dfrac{5ab^3}{-3c^2}\right)^2$；} & \xxt{$\left(\dfrac{2a^3y}{-x^2}\right)^3$；} \\
        \xxt{$\left(\dfrac{a+b}{4x}\right)^2$；} & \xxt{$\left(\dfrac{2x-y}{-5a^2}\right)^2$。}
    \end{tblr}

\end{xiaoxiaotis}

\xiaoti{下列各式对不对？如果不对，应怎样改正？}
\begin{xiaoxiaotis}

    \begin{tblr}{columns={18em, colsep=0pt}}
        \xxt{$\left(\dfrac{3b^2}{2a}\right)^3 = \dfrac{3b^6}{2a^3}$；} & \xxt{$\left(\dfrac{2x}{x+y}\right)^2 = \dfrac{4x^2}{x^2 + y^2}$。}
    \end{tblr}

\end{xiaoxiaotis}

\xiaoti{计算：}
\begin{xiaoxiaotis}

    \begin{tblr}{columns={18em, colsep=0pt}}
        \xxt{$\left(\dfrac{2x}{3y}\right)^2 \cdot \left(\dfrac{3y}{4x}\right)^3$；} & \xxt{$4x^2y \div \left(\dfrac{2x}{-y}\right)^2$；} \\
        \xxt{$\left(\dfrac{b^2}{ac}\right)^3 \div (-b^6c)$；} & \xxt{$\left(-\dfrac{a^2}{b}\right)^2 \cdot \left(-\dfrac{b^2}{a}\right)^3 \cdot \left(\dfrac{1}{ab}\right)^4$。}
    \end{tblr}

\end{xiaoxiaotis}

\end{xiaotis}

\end{enhancedline}

